Why was the Battle of Saratoga the turning point in the American Revolution?

In October 1777, the most important battle of the war occurred in Saratoga, New York. British forces under General Barry St. Leger (along with some Iroquois Indians who had allied with the British) and General John Burgoyne attempted to march across New York hoping to cut New England off from the rest of the states. American troops under General Horatio Gates defeated Burgoyne's army, inflicting 1,200 casualties and taking 7,000 prisoners. Saratoga was the first time that Americans troops had inflicted such large losses on the British, and so marked an important military turning point in the war.

Even more important, Saratoga also marked a major diplomatic turning point. After the American victory, France declared its support for the American Revolution and (four months later) declared war against the British. Until Saratoga, France had been unwilling to enter the war, expecting that the Americans would be defeated. French involvement challenged Britain's naval control over the Atlantic Ocean, threatened to stir resistance to British rule in Ireland, and posed a serious danger to British colonies in the Caribbean Sea. France did not play a major military role in most battles of the American Revolution, although French troops were an important part of the final Battle of Yorktown in 1781. Yet France's entry into the war added many new threats to Britain, forcing it to divert soldiers and sailors from the American war to defend its other interests.

In the winter of 1777-1778, Washington's army endured the miserable winter at Valley Forge, Pennsylvania, during which soldiers suffered from lack of food, inadequate clothing, poor shelter, and bitter cold. After the Battle of Monmouth, New Jersey, in 1778, most of the fighting in the Northern states ended, and the British turned their attention and hopes to the South.